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{\bf Edward A. Bender, E. Rodney Canfield, L. Bruce Richmond and
Herbert S. Wilf}
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{\bf A Discontinuity in the Distribution of Fixed Point Sums}
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The quantity $f(n,r)$, defined as the number of permutations of
the set $[n]=\{1,2,\dots n\}$ whose fixed points sum to $r$, shows a
sharp discontinuity in the neighborhood of $r=n$.
We explain this discontinuity and study the possible existence of
other discontinuities in $f(n,r)$ for permutations.
We generalize our results to other families of structures that
exhibit the same kind of discontinuities, by studying $f(n,r)$ when
``fixed points'' is replaced by  ``components of size 1'' in a
suitable graph of the structure.
Among the objects considered are permutations, all functions and
set partitions.

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